非光滑广义凸多目标规划的对偶

王彩玲,王泽升

(吉林大学 数学学院,长春 130012)

非光滑多目标优化理论应用广泛,而广义凸性在对偶理论中具有重要作用,已广泛用于解决优化问题.文献[1-6]给出了不同条件下广义凸函数的最优性条件与对偶理论.本文通过定义复合Q-ρ不变凸和S-δ不变凸函数,对其构成的不可微复合凸多目标规划问题给出了新的对偶理论,推广了文献[3-7] 的结果.设Q⊂p和S⊂m分别是具有内部的闭凸锥.

1 基本概念

考虑复合多目标规划问题(VP):

其中:g(G(x))=(g1(G1(x)),…,gm(Gm(x)))T;x∈X(X是Banach空间);f和g分别是n上实值局部Lipschitz向量函数;Fi和Gj:X→n分别是局部Lipschitz函数和Gateaux可微函数,它们的Gateaux导数分别记为(·)(i=1,2,…,p；j=1,2,…,m).记

Ω={x∈X|-g(G(x))∈S},

Q*和S*分别为Q和S的对偶锥,如Q*={λ∈p|λTv≥0,∀v∈Q}.

定义1如果存在ρ∈p,使得对∀及∀有

定义2如果存在δ∈m,使得对∀及∀有

其中U=W∩(-W).

2 对偶定理

考虑(VP)的Mond-Weir型对偶问题(VD):

定理1(弱对偶性) 设x和(y,Λ)分别是(VP)和(VD)的可行解,如果f(F)在y处关于函数η和θ：X×X→X是广义复合Q-ρ不变凸的,g(G)在y处关于函数η和θ是广义复合S-δ不变凸的,且ρ+Λδ∈Q,则

f(F(x))-f(F(y))∈W.

证明：由已知条件可得

由于对任意(VD)的可行解(y,Λ),由定理1可得

证明：由已知条件及定理1可知,对∀x∈Ω,有

⊂W.

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